dfferential equation with $df/dt = e^{\alpha f}$?

Question

Can anyone show me how to solve this kind of differential equation? $\frac{df}{dt}=e^{\alpha f}$, with$f(0) = 0$ and $Re(\alpha)<0$. Thank you very much!

Answer 1

Separable diff equation $$\frac{df}{dt}=e^{\alpha f}\\\frac{df}{e^{\alpha f}}=dt \\ \int_{0}^{t}\frac{df}{e^{\alpha f}} =\int_{0}^{t} ds \\ \int_{0}^{t}e^{-\alpha f}df =\int_{0}^{t} ds \\ e^{-\alpha f}|^t_0=s|^t_0\\ e^{-\alpha f(t)}-e^{-\alpha f(0)}=t \\e^{-\alpha f(t)}-e^{-\alpha (0)}=t\\ e^{-\alpha f(t)}-1=t\\ e^{-\alpha f(t)}=1+t\\ e^{+\alpha f(t)}=\dfrac{1}{1+t}\\$$